Physica A 266 (1999) 317–321
Optimal path in weak and strong disorder
Nehemia Schwartz a; ∗ , Markus Portoa;b , Shlomo Havlina , Armin Bundeb
a Minerva
b Institut
Center and Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel
fur Theoretische Physik III, Justus-Liebig-Universitat Giessen, Heinrich-Bu-Ring 16,
35392 Giessen, Germany
Abstract
We generate optimal paths between two given sites on a lattice representing a disordered
energy landscape by applying the Dijkstra algorithm. We study the geometrical and energetic
scaling properties of the optimal path under two dierent energy distributions that yield the weak
and strong disorder limits. Our numerical results, for both two and three dimensions, suggest that
the optimal paths in weak disorder are in the same universality class as the directed polymers
and in the strong disorder limit are fractals with exponents similar to that found by Cieplak et al.
c 1999 Published by Elsevier Science
(Phys. Rev. Lett. 72 (1994) 2320; 76 (1996) 3754). B.V. All rights reserved.
PACS: 61.43.Bn; 46.10.+z; 62.30.+d
Keywords: Optimal path; Dijkstra algorithm; Directed polymer
Optimal paths have been much of interest recently. The optimal path can be dened
as follows. Consider a d-dimensional lattice, where each bond is assigned with a random energy value taken from a given distribution. The optimal path between two sites
is dened as the path on which the sum of the energies is minimal. This problem is of
relevance to various elds, such as spin glasses [1], protein folding [2], paper rupture
[3], and traveling salesman problem [4]. Though much eort has been devoted to study
this problem, the general solution is still lacking. There exist two approaches developed recently to study this problem. Cieplak et al. [5] applied the max-ow algorithm
for a two-dimensional energy landscape. Another approach is to restrict the path to be
directed, that is, the path cannot turn backwards. This approach is the directed polymer
problem which has been extensively studied in the past years, see e.g., [6 –9].
In this manuscript, we adapt the Dijkstra algorithm from graph theory [10] for
generating the optimal path on a lattice with randomly distributed non-negative energies
assigned to the bonds [11]. This algorithm enables us to generate the optimal path
∗
Corresponding author.
c 1999 Published by Elsevier Science B.V. All rights reserved.
0378-4371/99/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 6 0 9 - 8
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N. Schwartz et al. / Physica A 266 (1999) 317–321
between any two sites on the lattice, not restricted to directed paths. We study the
geometrical and energetic properties of the optimal paths in d = 2 and 3 dimensions
using two dierent energy distributions. The rst distribution of energies is a uniform
distribution, P(E) = const. which gives the weak disorder limit, and the second distribution is P(E) ˙ 1=E which generates the strong disorder limit as will be explained
later. We calculate the scaling exponents for the width and the energy uctuations of
the optimal path in each case. For the distribution P(E) = const. we nd that for both
d = 2 and 3 the exponents are very close to those of directed polymers suggesting
that the non-directed optimal path (NDOP) is in the same universality class as directed
polymer (DP). Our results are in agreement with those found by Cieplak et al. [5] for
the two-dimensional case. This result indicates that in the case of uniformly distributed
energies the NDOPs are self ane and overhangs do not play an important role in the
geometry of NDOPs. For P(E) ˙ 1=E we nd that, for both d = 2 and 3, the optimal
paths are self similar and the exponents are very close to those found by Cieplak et al.
[13] who applied an alternative algorithm.
In the case of the uniform distribution we simulate both DPs and NDOPs on a
square lattice in the following way. Let x; y be the horizontal and vertical axes. We
choose the origin to be the source site and study the optimal paths connecting it with
all the sites on the line between [0; t] and [t; 0] for dierent values of t. In order
to compare NDOP to DP we dene the global optimal path as the minimal energy
path among all the paths with the same value of t. In Fig. 1 we compare a conguration of DP and NDOP on the same disordered energy landscape. It is seen that
in the NDOP only very few overhangs exist. To test the eect of the overhangs we
calculate the mean end-to-end distance R of the global optimal path as a function
of its length ‘. Our numerical results clearly indicate the asymptotic relation ‘ ∼ R
showing that the NDOPs are self-ane [8]. We study several properties, such as the
roughness exponent , the energy uctuation exponent for two and three dimensions, as well as the distribution of the endpoints of DP and NDOP (see Table 1
and Fig. 2). The above exponents are dened by the relations W ≡ hh2 i1=2 ∼ t and
E ≡ h(E − hEi)2 i1=2 ∼ t . Here, h is the transverse uctuation of the global optimal path which is distance between its endpoint and the line x = y; E is the energy
of the global optimal path which is the sum of all bond energies along the path.
The average is taken over dierent realizations. The generalization to three dimensions is straightforward. We nd that our results are independent of the distribution
interval.
As for the strong disorder limit, where the sum of energies along the path is governed
by one energy value, a uniform distribution is not suitable to reach this limit. Consider a
lattice, and an energy interval of (0; A), then most of the bonds will have a value which
is of the order of A for every A ¿ 0. Therefore, it will be impossible to connect two
sites in the optimal way and have the optimal path energies behave in a strong disorder
fashion. We therefore suggest an energy distribution P(log E)=const. (i.e. P(E) ˙ 1=E)
thus insuring that for dierent randomly generated integers will correspond two values
that dier at least by an order of magnitude. Let us consider now a lattice and the
N. Schwartz et al. / Physica A 266 (1999) 317–321
319
Fig. 1. The sets of all directed (the upper one) and non-directed (the lower one) optimal paths with t = 300
obtained for the same realization of quenched randomness in the lattice. The global optimal path which is
the minimal energy path among all the paths with the same t is shown by a thick line. In this particular
case the directed and non-directed global optimal paths do not overlap. In other cases they might overlap
signicantly though the rest of tree looks somewhat dierent.
proposed distribution taking the interval to be (0; ∞) then for every nite lattice which
has N bonds we are guaranteed to have N dierent values which dier from each
other at least by an order of magnitude thus insuring the existence of the optimal path
in the strong disorder limit. The eect of a cuto, Emax , on this distribution and the
crossover from the strong disorder to the weak disorder behavior have been studied
recently by Porto et al. [12]. Concentrating on the regime of the strong disorder, we
study the scaling properties of the optimal paths. We nd that dopt = 1:22 ± 0:02 in
d = 2 and dopt = 1:43 ± 0:03 in d = 3 where dopt is dened by ‘ ∼ Rdopt and ‘ is
the path’s length. These nding are in close agreement with the results obtained by
Cieplak et al. [13–16] which are dopt = 1:22 ± 0:01 for d = 2 and dopt = 1:42 ± 0:02
for d = 3.
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N. Schwartz et al. / Physica A 266 (1999) 317–321
Table 1
Width and energy uctuation exponents of DP and NDOP in two and three dimensions using uniform
distribution. The error bar were estimated from taking ve ensembles of 104 congurations each for
d = 3 and 500 congurations each for d = 3
d=2
d=3
DP
NDOP
DP
NDOP
0:66 ± 0:02
0:32 ± 0:02
0:67 ± 0:02
0:32 ± 0:02
0:60 ± 0:05
0:19 ± 0:07
0:63 ± 0:05
0:18 ± 0:07
Fig. 2. Scaling of the disribution P(h; t) of the endpoints in (a) DP and (b) NDOP. The dierent symbols
represent dierent values of t between t = 10 and 300.
N. Schwartz et al. / Physica A 266 (1999) 317–321
321
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